#500924

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,7,4,6)(9,10,13,19)(11,21,24,16)(12,28,29,18)(14,17,31,38)(15,33,30,20)(22,36,25,40)(23,34,27,37)(26,35,32,39);(1,3,5,4)(2,6,8,7)(9,11,13,24)(10,16,19,21)(12,23,29,27)(14,32,31,26)(15,22,30,25)(17,35,38,39)(18,34,28,37)(20,36,33,40);(10,17)(11,22)(12,29)(14,30)(15,31)(16,34)(18,20)(19,38)(21,37)(23,26)(24,25)(27,32)(28,33)(36,40);(1,4,5,3)(2,7,8,6)(9,12,13,29)(10,18,19,28)(11,23,24,27)(14,30,31,15)(16,34,21,37)(17,33,38,20)(22,32,25,26)(35,40,39,36)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=di,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319933, C₂×#500924
maximal normal subgroup of	#72319933, C₂×#500924
maximal quotient of	C₂×#500924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408120);